Noise sensitivity of the minimum spanning tree of the complete graph

Omer Israeli, Yuval Peled*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study the noise sensitivity of the minimum spanning tree (MST) of the <![CDATA[ $n$ ]]> -vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by <![CDATA[ $n^{1/3}$ ]]> and vertices are given a uniform measure, the MST converges in distribution in the Gromov-Hausdorff-Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability <![CDATA[ $\varepsilon \gg n^{-1/3}$ ]]>, then the pair of rescaled minimum spanning trees - before and after the noise - converges in distribution to independent random spaces. Conversely, if <![CDATA[ $\varepsilon \ll n^{-1/3}$ ]]>, the GHP distance between the rescaled trees goes to <![CDATA[ $0$ ]]> in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of <![CDATA[ $n^{-1/3}$ ]]> coincides with the critical window of the Erd 's-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.

Original languageEnglish
JournalCombinatorics Probability and Computing
DOIs
StateAccepted/In press - 2024

Bibliographical note

Publisher Copyright:
© The Author(s), 2024. Published by Cambridge University Press.

Keywords

  • Keywords:
  • minimum spanning tree
  • noise sensitivity
  • Random graphs
  • scaling limits

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